Question

A brewery needs to purchase glass bottles that can withstand an internal pressure of at least 150 pounds per square inch (psi). A prospective bottle vendor claims its production process yields bottles with a mean strength of 157 psi and a standard deviation of 3 psi.

a. Assume that the strength of the vendor’s bottles is normally distributed. Calculate the probability that a single bottle chosen randomly from this vendor’s factory would fail to meet the brewer’s standard, assuming the vendor’s claim is true.

b.         The brewer tests a sample of 40 bottles from this vendor and finds the mean strength of the sample to be 155.7. Assuming the vendor’s strength claim to be true, what is the probability of obtaining such a sample with a mean this far or farther below the claimed mean? What does this answer suggest about the veracity of the vendor’s claim?

c.          Which of the following changes would make observing a sample as described in part b. more likely:

            i)          Reducing the claimed mean to 156, holding the standard deviation at 3; or

            ii)         Reducing the standard deviation to 2, holding the mean at 157?

            [Note: You do not have to calculate new probabilities to answer this. But be sure you explain the reasoning behind your answers fully using the appropriate formulas]

Answer 1

a) Assuming that the strength of the vendor’s bottles is normally distributed, then

random variable X is normally distributed with

a single bottle chosen randomly from this vendor’s factory would fail to meet the brewer’s standard if the glass bottles can withstand an internal pressure of less than 150 pounds per square inch (psi)

Hence the required probability that thhe probability that a single bottle chosen randomly from this vendor’s factory would fail to meet the brewer’s standard, assuming the vendor’s claim is true is

Converting to standard Z score

b) sample info:

the probability of obtaining such a sample with a mean this far or farther below the claimed mean

c) the standard normal distribution is symmetric about zero

option i)reducing the claimed mean to 156 will make the standard score equal to

this will be 2 standard deviation away from the mean

option ii) reducing the standard deviation to 2 will make the standard Z score

this will be more than 3 standard deviation away from the mean

hence option i is right